A priori bounds for stochastic porous media equations via regularity structures

2025/07/04 by Markus Tempelmayr, Tempelmayr, Markus, Hendrik Weber +1
Economics, Econometrics and Finance · Mathematics · #35K65 #60H17 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Navier-Stokes equation solutions #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2507.03575

openalex publication_date 2025/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a priori bounds for solutions of singular stochastic porous media equations with multiplicative noise in their natural L¹-based regularity class. We consider the first singular regime, i.e.~noise of space-time regularity α-2 for α∈(2/3,1), and prove modelledness of the solution in the sense of regularity structures with respect to the solution of the corresponding linear stochastic heat equation. The proof relies on the kinetic formulation of the equation and a novel renormalized energy inequality. A careful analysis allows to balance the degeneracy of the diffusion coefficient against sufficiently strong damping of the multiplicative noise for small values of the solution.

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